Some characterizations of the complex projective space via Ehrhart polynomials

Let $P_{\lambda {\mathrm{\Sigma }}_n}$ be the Ehrhart polynomial associated to an intergal multiple $\lambda$ of the standard symplex ${\mathrm{\Sigma }}_n\subset {\mathbb{R}}^n$. In this paper we prove that if $(M,L)$ is an $n$-dimensional polarized toric manifold with associated Delzant polytope $\mathrm{\Delta }$ and Ehrhart polynomial $P_{\mathrm{\Delta }}$ such that $P_{\mathrm{\Delta }}=P_{\lambda {\mathrm{\Sigma }}_n}$, for some $\lambda \in {\mathbb{Z}}^{+}$, then $(M,L)\cong (\mathbb{C}{P}^n,O(\lambda ))$ (where $O(1)$ is the hyperplane bundle on $\mathbb{C}{P}^n$) in the following three cases: 1. arbitrary $n$ and $\lambda =1$, 2. $n=2$ and $\lambda =3$, 3. $\lambda =n+1$ under the assumption that the polarization $L$ is asymptotically Chow semistable.